9.5 Integrating Vector-Valued Functions

• What is a vector-valued function?

  A function that outputs a vector
• Each component function of a vector-valued function represents the x, y, and z coordinates of the vector as a function of the parameter t.


  True
• Understanding the components of a vector-valued function is essential for interpreting its geometric meaning and behavior.


  True
• What does the definite integral of the x-component, $\int_{a}^{b} x(t) dt$, represent in the context of a vector-valued function?


  The area under $x(t)$ from $a$ to $b$
• Each component of a vector-valued function represents the position of the vector in the x, y, and z directions respectively.
• How is the definite integral of a vector-valued function computed?

  Integrating each component separately
• The component functions of a vector-valued function depend on the parameter $t$.


  True
• A vector-valued function outputs a vector rather than a scalar.
• What does the $y(t)$ component of a vector-valued function represent?


  Position in the y-direction
• The definite integral of a vector-valued function $\vec{r}(t) =$$\langle x(t), y(t), z(t) \rangle$ from $a$ to $b$ is a vector
• How is the definite integral of a vector-valued function computed?

  Component-wise integration
• The definite integral of $\vec{r}(t) =$$\langle t, t^{2} \rangle$ from 0 to 2 is $\langle 2, \frac{8}{3} \rangle$.


  True
• What is the first step in integrating a vector-valued function?

  Identify components
• The indefinite integral of $\vec{r}(t) =$$\langle t, t^{2}, e^{t} \rangle$ is \left\langle \frac{1}{2}t^{2} +$C_{1}, \frac{1}{3}t^{3} +$$C_{2}, e^{t} +$ C_{3} \right\rangle.


  True
• The arc length of $\vec{v}(t) =$$\langle 3\cos t, 3\sin t, 4t \rangle$ from 0 to 2π is 10π
• A vector-valued function can be represented as $\vec{r}(t) =$$\langle x(t), y(t), z(t)\rangle$, where $x(t)$, $y(t)$, and $z(t)$ are called the component functions.
• Match the component of a vector-valued function with its interpretation:

  $x(t)$ ↔️ Position in the x-direction

  $y(t)$ ↔️ Position in the y-direction
• The definite integral of a vector-valued function is computed by integrating each component function separately.
• Match the component of a vector-valued function with its corresponding definite integral:

  $x(t)$ ↔️ $\int_{a}^{b} x(t) dt$

  $y(t)$ ↔️ $\int_{a}^{b} y(t) dt$

  $z(t)$ ↔️ $\int_{a}^{b} z(t) dt$
• How is the definite integral of a vector-valued function \vec{r}(t) = \langle x(t), y(t), z(t) \rangle</latex> calculated from $a$ to $b$?


  By integrating each component separately
• The definite integral of a vector-valued function $\vec{r}(t) =$$\langle x(t), y(t), z(t) \rangle$ from $a$ to $b$ is given by \left\langle \int_{a}^{b} x(t) dt, \int_{a}^{b} y(t) dt, \int_{a}^{b} z(t) dt \right\rangle</latex>
• The variables $x(t)$, $y(t)$, and $z(t)$ are the component functions of the vector-valued function $\vec{r}(t)$.


  True
• Steps for integrating a vector-valued function $\vec{r}(t) =$$\langle x(t), y(t), z(t) \rangle$.


  1️⃣ Identify the component functions: $x(t)$, $y(t)$, and $z(t)$.

  2️⃣ Integrate each component: $\int x(t) dt$, $\int y(t) dt$, $\int z(t) dt$.

  3️⃣ Write the result as a vector: $\int \vec{r}(t) dt =$ \left\langle F_{x}(t) +$C_{x}, F_{y}(t) +$$C_{y}, F_{z}(t) +$ C_{z} \right\rangle.
• What does a vector-valued function output?

  A vector
• A vector-valued function can be represented as \vec{r}(t) = \langle x(t), y(t), z(t)\rangle</latex>
• Match the component of a vector-valued function with its interpretation:

  $x(t)$ ↔️ Position in the x-direction

  $y(t)$ ↔️ Position in the y-direction

  $z(t)$ ↔️ Position in the z-direction
• The component functions of a vector-valued function represent the x, y, and z coordinates as functions of the parameter $t$.


  True
• Match the component of a vector-valued function with its corresponding definite integral:

  $x(t)$ ↔️ $\int_{a}^{b} x(t) dt$

  $y(t)$ ↔️ $\int_{a}^{b} y(t) dt$

  $z(t)$ ↔️ $\int_{a}^{b} z(t) dt$
• The definite integral of the $x(t)$ component is \int_{a}^{b} x(t) dt</latex>.


  True
• The definite integral of a vector-valued function \vec{r}(t)</latex> from $a$ to $b$ is represented as a vector
• Match the property of definite integrals with its description:

  Linearity Property ↔️ $\int_{a}^{b} [\vec{r}(t) +$$\vec{s}(t)] dt =$$\int_{a}^{b} \vec{r}(t) dt +$$\int_{a}^{b} \vec{s}(t) dt$

  Constant Multiple Property ↔️ $\int_{a}^{b} k\vec{r}(t) dt =$$k \int_{a}^{b} \vec{r}(t) dt$
• When integrating a vector-valued function, each component $f(t)$ is integrated as \int f(t) dt = F(t) + C
• What does the definite integral of velocity calculate?

  Displacement